Coordinate system
- Cartesian system $(\vec i,\vec j,\vec k)$
- orthogonal: $\vec i\cdot\vec j=\vec j\cdot\vec k=\vec k\cdot\vec i=0$
- right-handed screw relation: $\vec i\times\vec j=\vec k$, $\vec j\times\vec k=\vec i$, $\vec k\times \vec i=\vec j$
- $A\cdot B = A_1B_1+A_2B_2+A_3B_3$
- $$
A\times B=\det(\left[
\begin{matrix}
\vec i & \vec j & \vec k\newline
A_1 & A_2 & A_3\newline
B_1 & B_2 & B_3
\end{matrix}
\right])
$$
- polar coordinate systems $(\vec e_\rho, \vec e_\phi)$
- moving frames
- position vector $\vec\rho=\rho\vec e_\rho$
- $\vec e_\rho=cos\varphi\vec i+sin\varphi\vec j$
- $\frac{d\vec e_\rho}{dt}=\dot{\varphi}\vec e_\varphi$
- $\frac{d\vec e_\varphi}{dt}=-\dot{\varphi}\vec e_\rho$
- Spherical systems $(\vec e_\rho, \vec e_\phi, \vec k)$
- Cylindrical systems
- intrinsic system
temp
- In sperical coordinate system: $\nabla^2=\frac{1}{r^2}\frac{\partial}{\partial r}(r^2\frac{\partial}{\partial r})+\frac{1}{r^2}[\frac{1}{sin\theta}\frac{\partial}{\partial \theta}(sin\theta\frac{\partial}{\partial \theta})+\frac{1}{sin^2\theta}\frac{\partial^2}{\partial \varphi^2}]$
场论
- $\nabla\times(\nabla\times A) = \nabla(\nabla\cdot A) - \nabla^2 A$
- 柱坐标系： $\nabla = \frac{\partial}{\partial\rho}+\frac{1}{\rho}\cdot\frac{\partial}{\partial\phi}+\frac{\partial}{\partial z}$